Molecules in solution are generally characterized by their weight averaged molar mass M, their mean square radius <rg2>=∫r2dm/∫dm (here r is the distance from the center of mass of the molecule and dm the mass of a small volume at that distance), and the second virial coefficient A2. In some cases, the third virial coefficient A3 is also of interest. In other cases, the cross-virial coefficient A2AB between two distinct molecules A and B is of interest. M and <rg2> are properties of the individual molecules, averaged over all the molecules in the solution; the virial coefficients are a measure of the average interaction between the molecules as mediated by the solvent. For unfractionated solutions, these properties may be determined by measuring the manner by which they scatter light using the method described by Bruno Zimm in his seminal 1948 paper which appeared in the Journal of Chemical Physics, volume 16pages 1093 through 1099. The light scattered from a small volume of the solution is measured over a range of angles and concentrations. The collection of light scattering data over a range of scattering angles is referred to more commonly as multiangle light scattering, MALS. The properties derived from the light scattering measurements for a single type of molecule are related through the formula developed by Zimm and corrected by W. A. J. Bryce in Polymer 10 804-809 (1969):R*(c,θ)=McP(θ)−2A2[MP(θ)c]2−[3A3Q(θ)−4A22MP2(θ)][MP(θ)]2c3   (1)where R*(c,θ)=R(c,θ)/K*, R(c,θ) is the measured excess Rayleigh ratio in the direction θ per unit solid angle defined as R(θ)=[Is(θ)−Isolv(θ)]r2/[I0V], Is(θ) is the intensity of light scattered by the solution a function of angle, Isolv(θ) is the intensity of light scattered from the solvent as a function of angle, I0 is the incident intensity, r is the distance from the scattering volume to the detector, V is the illuminated volume seen by the detectors, K=4π2 n02/(NAλ04), and K*=K(dn/dc)2, NA is Avogadro's number, dn/dc is the refractive index increment, n0 is the solvent refractive index, and λ0 is the wavelength of the incident light in vacuum. P(θ) is the form factor of the scattering molecules defined as P(θ)=limc→0 R(θ)/R(0). The general form of P(θ) was derived by P. Debye in J. Phys. Colloid Chem., 51 18 (1947) as:
                              P          ⁡                      (            θ            )                          =                  1          -                                    P              1                        ⁢                                          sin                2                            ⁡                              (                                  θ                  2                                )                                              +                                    P              2                        ⁢                                          sin                4                            ⁡                              (                                  θ                  2                                )                                              +          …                                    (        2        )            P1 is related to the mean square radius via
            P      1        =                                        (                          2              ⁢              k                        )                    2                3            ⁢              〈                  r          g          2                〉              ,and k=2π/λ with λ being the wavelength of the incident light in the solvent. P2 is related to the mean square radius <rg2> and the mean quadri-radius <rg4> via
      P    2    =                    2        ⁢                  k          4                    45        ⁢                  (                              10            ⁢                                          〈                                  r                  g                  2                                〉                            2                                -                      3            ⁢                          〈                              r                g                4                            〉                                      )            .      This equation is an approximation based on a series expansion in powers of concentration and sin2(θ/2); as such, the degree of accuracy depends on the relative magnitudes of the higher order terms.
The standard method, also known as the “plateau method,” of carrying out this measurement involves preparing a series of samples with increasing concentrations of known values; sequentially introducing the samples to a MALS detector, whether by inserting glass vials containing the samples in the light beam or by injecting the samples into a flow cell located in the beam; acquiring the scattered intensity at each angle by means of a photodetector; calculating the excess Rayleigh ratios for each concentration and angle; and fitting the data to Eq. (1) to extract M, <rg2>, A2 and A3.
In the case of injecting the sample into a flow cell containing some previous sample or solution, sufficient volume must be injected in order to “saturate” the cell, i.e. to bring the concentration in the cell to the original, known sample concentration; this may be accomplished by observing the scattering signal and flowing the sample until the value of the signal vs. time reaches a plateau, which occurs asymptotically with an exponential dependence. Alternatively, a concentration detector with a flow cell may be added to the flow path, and sufficient sample must be injected in order for both the MALS and concentration signals attain a plateau over time; in this manner the concentration in the MALS detector may be inferred from the concentration in the concentration detector, to provide the values of R and c in Eq. (1). Typical volumes per injection required to saturate the flow cells are 1-3 mL.
Recognizing the asymptotic approach to the correct concentration values in each cell, a more precise measurement may be obtained by following an increasing concentration series with a decreasing series, where the correct concentrations are asymptotically approached from above rather than from below. The average of the two measurements (increasing and decreasing concentrations) provides a more accurate calculation, at the cost of double the total sample and double the measurement time. Following U.S. Pat. No. 6,651,009 by Trainoff et al., the '009 patent, one may define figures of merit, FOM, describing the magnitude of the A2 and A3 terms in the virial expansion of the light scattering equation, relative to the pure mass term; these are readily derived from Eq. (1) as 2A2Mc−4A22M2c2 and 3A3Mc2, respectively. The assumption that the light scattering may be described by such a virial expansion implies convergence of the equation, i.e. the magnitudes of successively higher-order terms drop off fairly quickly; in other words 1>>FOM(A2)>>FOM(A3), and the smaller the figures of merit, the better the approximation. On the other hand, it is clear from considerations of signal-to-noise that the figures of merit must be of some finite value in order to obtain a reliable measurement. For a particular sample and instrument, the desirable upper and lower limits of the FOM for determining A2 and A3 are set by these considerations.
In U.S. Pat. No. 6,411,383Wyatt describes a related method for measuring M, <rg2>, and A2, using a single injection of unfractionated sample of finite volume, flowing through a MALS detector and a concentration detector. The sample injection is preceded and followed by sufficient pure solvent to bring the MALS and concentration signals back to baseline (Isolv) and is denoted a “peak” in the signal. Because of the finite nature of the sample, upon flowing through the system it is diluted and broadened so that different parts of the injection present different concentrations. Instead of applying Eq, (1) to multiple injections with a single value of I(θ) and c per injection, the inventor calculates the sums of I(θ), c and c2 over the single peak, and determines <rg2> and A2 via Eq. (1) and a priori knowledge of M.
This method, utilizing a single flowing peak, is denoted herein as the “Wyatt peak” method. Since flow cell saturation is not required, a much smaller volume is required, typically in the range of 200-500 μL, where the maximum elution interval concentration corresponds to that which would be attained with the plateau method.
In passing from detector to detector, the sample peak is further broadened and reduced in height due to mixing and dilution. Hence the time-dependent concentration signal from the concentration detector does not precisely replicate the time-dependent concentration present in the light scattering detector. Since the light scattering signal is not a linear function of concentration, applying Eq. (1) to the sequential values of the light scattering and concentration leads to some degree of error in calculating the virial coefficients. The error in the Wyatt peak method increases both as the interdetector broadening increases, and as the deviation of the MALS signal from linear dependence on concentration increases, i.e. with larger FOM.
The '009 patent describes a correction factor intended to reduce this error, denoted herein the “Trainoff-Wyatt peak method”. In this method, a series of peaks of different concentrations are injected into the detectors. Typical volumes of these injections are 100-200 μL. In the absence of interdetector broadening, it is possible to calculate the parameters of interest by summing R(θ), c and c2 over each peak and fitting the sums to Eq. (1). Again, the broadening effect introduces errors into the calculation. The '009 patent shows that, in the case that the broadening causes only small changes to the widths of the peak, the error can be corrected by a simple multiplicative factor. This factor can be determined in several ways. The simplest is a calibration method in which the results of the injection method are compared to that of the plateau method for a reference standard. The correction factor determined thereby can be used for subsequent unknown samples. In the special case of Gaussian peaks, the correction factor can be inferred from the ratio of the peak widths. The limitation of this method is that it does not work well for large broadening, and it requires extra effort to determine the correction factor a priori.
Therefore, a method that employs flowing peaks for minimal sample quantities as in the '009 patent, but analyzes the peak data for M, <rg2>, A2 and A3 in a manner independent of inter-detector broadening effects, without restriction on the peak shape, would be advantageous.
Cross virial coefficients, measuring the interactions between different species of molecules A and B, are also quantities of fundamental importance in science and industry. The light scattering equation for a solution of two molecular species is presented in Eq. (3), to first order. As for the single-species case, the self- and cross-virial coefficients may be measured with a series of injections containing different concentrations, in this case of both species, fitting the light scattering and concentration values to the equation, analogous to the plateau method for MA, MB, <rg2>A, <rg2>B, A2A, A2B and A2AB where A2A and A2 may be considered “self-virial coefficients”, and A2AB the cross virial coefficient. A method for characterizing cross-virial coefficients using minimal sample quantities, akin to the Wyatt peak method for self-virial coefficients, would be advantageous.
                                          R            ⁡                          (                                                c                  A                                ,                                  c                  B                                ,                θ                            )                                K                =                                                            (                                                      ⅆ                    n                                                        ⅆ                                          c                      A                                                                      )                            2                        ⁢                          {                                                                    M                    A                                    ⁢                                      c                    A                                    ⁢                                                            P                      A                                        ⁡                                          (                      θ                      )                                                                      -                                  2                  ⁢                                                                                    A                        2                        A                                            ⁡                                              [                                                                              M                            A                                                    ⁢                                                                                    P                              A                                                        ⁡                                                          (                              θ                              )                                                                                ⁢                                                      c                            A                                                                          ]                                                              2                                                              }                                +                                                    (                                                      ⅆ                    n                                                        ⅆ                                          c                      B                                                                      )                            2                        ⁢                          {                                                                    M                    B                                    ⁢                                      c                    B                                    ⁢                                                            P                      B                                        ⁡                                          (                      θ                      )                                                                      -                                  2                  ⁢                                                                                    A                        2                        B                                            ⁡                                              [                                                                              M                            B                                                    ⁢                                                                                    P                              B                                                        ⁡                                                          (                              θ                              )                                                                                ⁢                                                      c                            B                                                                          ]                                                              2                                                              }                                -                                    (                                                                    ⅆ                    n                                                        ⅆ                                          c                      A                                                                      -                                                      ⅆ                    n                                                        ⅆ                                          c                      B                                                                                  )                        ⁢            4            ⁢                          A              2              AB                        ⁢                          M              A                        ⁢                          M              B                        ⁢                                          P                A                            ⁡                              (                θ                )                                      ⁢                                          P                B                            ⁡                              (                θ                )                                      ⁢                          c              A                        ⁢                                          c                B                            .                                                          (        3        )            
Various types of on-line concentration detectors are known, including UV-visible absorbance, fluorescence, and differential refractive index, dRI, detectors. dRI detectors are particularly useful in combination with light scattering measurements, and are sufficiently versatile to measure a wide range of soluble macromolecules. One drawback of the dRI measurement is the necessity for completely dialyzing protein samples against the solvent, for the very reason that the dRI detector is sensitive to the salts and excipients, as well as dissolved gasses, that may be present in the protein sample but not in the pure buffer.
Dialysis is also important in the virial coefficient measurement because the virial coefficients vary with buffer conditions, and these must be well-defined for the measurement to be meaningful. In some instances, the effect of the buffer on the molecular interactions is determined by measuring virial coefficients under several different buffers, and aliquots of the same sample must be dialyzed against each buffer. However, dialysis can be a tedious and time-consuming process, and a means of combining the flowing peak measurement with in-line dialysis would be advantageous in automating these measurements.